Lengths, Widths, Surfaces

by Hoyrup, Jens

• ISBN13: 9780387953038
• ISBN10: 0387953035
• Format: Hardcover
• Copyright: 2002-03-01
• Publisher: Springer Verlag

Summary

In this new examination of Babylonian cuneiform texts, Jens Høyrup proposes a new interpretation, based on the fact that the tablets are almost entirely students' workbooks. The knowledge of mathematics expressed in these tablets is entirely practical, for use in surveying, accounting, and building, rather than theoretical. Høyrup argues that the notion of algebraic manipulation, like other parts of a theoretical mathematics is indeed a later invention.

Table of Contents

Preface	vii
Introduction	1	(10)
The Discovery of Babylonian ``Algebra''	1	(2)
The Standard Interpretation	3	(5)
The Texts, the Genre, and the Problems	8	(3)
A New Reading	11	(39)
An Example	11	(3)
Structural Analysis and Close Reading	14	(1)
Numbers and Measures	15	(3)
Mathematical Operations	18	(14)
Additive Operations	19	(1)
Subtractive Operations	20	(1)
``Multiplications''	21	(2)
Rectangularization, Squaring, and ``Square Root''	23	(4)
Division, Parts, and the igi	27	(4)
Bisection	31	(1)
Mathematical Organization and Metalanguage	32	(8)
The Standard Format of Problems	32	(1)
Standard Names and Standard Representation	33	(4)
Structuration	37	(2)
Recording	39	(1)
The ``Conformal Translation''	40	(10)
Table 1: Akkadian Terms and Logograms with Appurtenant Standard Translation	43	(4)
Table 2: The Standard Translations with Akkadian and Logographic Equivalents	47	(3)
Select Textual Examples	50	(46)
BM 13901 #1	50	(2)
BM 13901 #2	52	(1)
BM 13901 #3	53	(2)
YBC 6967	55	(3)
BM 13901 #10	58	(2)
BM 15285 #24	60	(1)
VAT 8390 #1	61	(4)
YBC 6295	65	(1)
BM 13901 #8-9	66	(5)
BM 13901 #12	71	(2)
BM 13901 #14	73	(4)
VAT 8389 #1	77	(5)
VAT 8391 #3	82	(3)
TMS XVI	85	(4)
TMS IX	89	(7)
Methods	96	(12)
``Naive'' Cut-and-Paste Geometry	96	(3)
Scaling and Other Changes of Variable	99	(1)
Accounting, Coefficients, Contributions	100	(1)
Single (and Other) False Positions - and Bundling	101	(2)
Drawings? Manifest or Mental Geometry?	103	(5)
Further ``Algebraic'' Texts	108	(119)
BM 13901 #18	108	(3)
YBC 4714	111	(26)
#1-#3	132	(1)
#4-7, 10-12	133	(1)
#8-9	133	(1)
#13-20	133	(1)
#21-28	134	(1)
#29	134	(1)
#30-39	135	(1)
General Commentary	136	(1)
BM 85200 + VAT 6599	137	(25)
The Third-Degree Problems	149	(5)
The second degree: Length-Width, Depth-Width, and Length-Depth	154	(4)
Second-Degree igum-igibum-Problems	158	(1)
First-Degree Problems	159	(2)
Clues to Teaching Methods	161	(1)
AO 8862 #1-4	162	(12)
#1	169	(1)
#2	170	(1)
#3	171	(1)
Average and Deviation	172	(2)
#4	174	(1)
YBC 6504	174	(5)
AO 6770 #1	179	(2)
TMS VII	181	(7)
#1	185	(1)
#2	186	(2)
A Concluding General Observation	188	(1)
TMS VIII	188	(6)
#1	191	(2)
#2	193	(1)
TMS XIX	194	(6)
#1	197	(1)
#2	197	(3)
YBC 4668, Sequence C, #34, #38-53	200	(3)
YBC 4713 #1-8	203	(3)
TMS XIII	206	(3)
VAT 7532	209	(4)
IM 52301 #2	213	(4)
BM 85194 #25-26	217	(5)
#26	220	(1)
#25	220	(2)
BM 13901 #23	222	(5)
Quasi-Algebraic Geometry	227	(51)
Introductory Considerations	227	(4)
Angles and Similarity	227	(1)
Perpendicularity and Orientation	228	(1)
Rectangles, Triangles, Trapezia, and ``Surveyors' Formula''	229	(2)
IM 55357	231	(3)
VAT 8512	234	(5)
Str 367	239	(5)
YBC 4675	244	(6)
UET V, 864	250	(4)
YBC 8633	254	(3)
Db2-146	257	(4)
YBC 7289	261	(4)
TMS I	265	(3)
VAT 6598 #6-7	268	(4)
BM 85194 #20-21	272	(3)
BM 85196 #9	275	(1)
Summary Observations	276	(2)
Old Babylonian ``Algebra'': A Global Characterization	278	(31)
Algebra?	278	(4)
Equations	282	(1)
Distinctive Characteristics	282	(10)
The Given and the Merely Known	283	(1)
``Pedantic Repetitiveness''	284	(1)
Favourite Configurations	285	(1)
Favourite Problems	286	(1)
``Remarkable Numbers''	287	(4)
``Broad Lines'' and ``Thick Surfaces''	291	(1)
Did They ``Know'' It?	292	(7)
Zero	293	(1)
Negative Numbers	294	(3)
Irrational Numbers?	297	(1)
Logograms as ``Mathematical Symbols''?	298	(1)
Overall Organization	299	(10)
Technical Terminology	299	(3)
Mathematics?	302	(7)
The Historical Framework	309	(8)
Landscape and Periodization	309	(2)
Scribes, Administration - and Mathematics	311	(6)
The ``Finer Structure'' of the Old Babylonian Corpus	317	(45)
Description of the Groups	319	(39)
Group 7: The Eshnunna Texts	319	(7)
Group 8: The Susa Texts	326	(3)
Groups 6 and 5: Goetze's ``Northern'' Groups	329	(4)
Groups 4 and 3: Goetze's ``Uruk Groups''	333	(4)
Group 1: The ``Larsa'' Group	337	(8)
Group 2 - a Non-Group?	345	(4)
The Series Texts	349	(3)
Old Babylonian Ur and Nippur	352	(6)
Summarizing	358	(1)
The Outcome	358	(4)
The Origin and Transformations of Old Babylonian Algebra	362	(38)
Practitioners' Knowledge and Specialists' Riddles	362	(6)
A Long and Widely Branched Tradition: the Lay Surveyors	368	(7)
The Sumerian School: the Vocabulary as Evidence	375	(3)
The ``Surveyors' Proto-algebra''	378	(2)
Scholastization	380	(5)
An Aside on the Pythagorean Rule	385	(2)
The Later Phases	387	(2)
Seleucid Procedure Texts	389	(2)
BM 34568	391	(9)
Repercussions and Influences	400	(18)
Greek Theoretical Mathematics	400	(5)
Demotic Egypt	405	(1)
Greek Underbrush	406	(2)
India	408	(2)
Impact in Islamic and Post-Islamic Mathematics: Towards Early Modern Algebra	410	(8)
Abbreviations and Bibliography	418	(10)
Index of Tablets	428	(6)
Index of Akkadian and Sumerian Terms and Key Phrases	434	(6)
Name Index	440	(3)
Subject Index	443